Let $A,B\in\mathcal{M}_n(\mathbb{Z})$ and $p$ a prime number. Prove that $$\text{Tr}((A+B)^p)\equiv\text{Tr}(A^p+B^p)\pmod p.$$

a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$. b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.

Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers. Calculate $\det(A)$.

Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $ Prove that $ 0=\det\left( A+\text{adj} A \right) . $ [i]Neculai Moraru[/i]

A matrix $A=(a_{ij})$ is called [i]nice[/i], if it has the following properties: (i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$; (ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$; (iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$, then either $i=k$ or $j=\ell$; (iv) for each $s=1,2,\dots,2t-1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$. Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrixes is even. For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1\\2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3\\2 & 4 & 4 \end{pmatrix}$.

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

On a chess table $n\times m$ we call a [i]move [/i] the following succesion of operations (i) choosing some unmarked squares, any two not lying on the same row or column; (ii) marking them with 1; (iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move). We call a [i]game [/i]a succession of moves that end in the moment that we cannot make any more moves. What is the maximum possible sum of the numbers on the table at the end of a game?

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?

Let $A$ be a real $n\times n$ matrix such that $A^3=0$ a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation $X+AX+XA^2=A$ b) Express $X$ in terms of $A$

We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.

Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$

Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$. [i]J. Szucs[/i]

Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.

Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible. Prove that $ A^5\neq I_5. $

Given the determinant of order $n$ $$\begin{vmatrix} 8 & 3 & 3 & \dots & 3 \\ 3 & 8 & 3 & \dots & 3 \\ 3 & 3 & 8 & \dots & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & 3 & 3 & \dots & 8 \end{vmatrix}$$ Calculate its value and determine for which values of $n$ this value is a multiple of $10$.

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Answer wither (i) or (ii): (i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$ (ii) Let $(a_{ij})$ be a matrix such that $$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$ for all $i.$ Show that the determinant is not equal to $0.$

A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that: (i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and (ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$ Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$